Transfer the reserved oil to a large skillet over medium heat. Add the asparagus and cook until bright green and slightly tender, about 2 minutes. Add the mushrooms and saute until slightly softened, about 2 minutes. Add the tomatoes, paprika, fennel and salt to taste. Cook until the tomatoes are heated through, about 2 minutes longer.
Heat an omelet pan or small nonstick saute pan over medium-low heat. Add the remaining 2 tablespoon of oil. Pour in the eggs and cook until the bottom is set but there is still a pool of uncooked eggs on top, 2 to 3 minutes. Lift up the bottom of the omelet with a spatula, allowing the uncooked eggs to flow back into the pan. Quickly add the cooked vegetables and goat cheese to the uncooked egg and invert the cooked egg over the vegetables. Cook until the eggs in the bottom of the pan are just set, about another 2 minutes. Transfer to a plate and garnish with the fried basil leaves.